# Lesson 12

Meaning of Exponents

Let’s see how exponents show repeated multiplication.

### 12.1: Notice and Wonder: Dots and Lines

What do you notice? What do you wonder?

### 12.2: The Genie’s Offer

You find a brass bottle that looks really old. When you rub some dirt off of the bottle, a genie appears! The genie offers you a reward. You must choose one:

• $50,000; or • A magical$1 coin. The coin will turn into two coins on the first day. The two coins will turn into four coins on the second day. The four coins will double to 8 coins on the third day. The genie explains the doubling will continue for 28 days.
1. The number of coins on the third day will be $$2 \boldcdot 2 \boldcdot 2$$. Write an equivalent expression using exponents.
2. What do $$2^5$$ and $$2^6$$ represent in this situation? Evaluate $$2^5$$ and $$2^6$$ without a calculator.
3. How many days would it take for the number of magical coins to exceed \$50,000?
4. Will the value of the magical coins exceed a million dollars within the 28 days? Explain or show your reasoning.

Explore the applet. (Why do you think it stops?)

A scientist is growing a colony of bacteria in a petri dish. She knows that the bacteria are growing and that the number of bacteria doubles every hour.

When she leaves the lab at 5 p.m., there are 100 bacteria in the dish. When she comes back the next morning at 9 a.m., the dish is completely full of bacteria. At what time was the dish half full?

### 12.3: Make 81

1. Here are some expressions. All but one of them equals 16. Find the one that is not equal to 16 and explain how you know.

$$2^3\boldcdot 2$$

$$4^2$$

$$\frac{2^5}{2}$$

$$8^2$$

2. Write three expressions containing exponents so that each expression equals 81.

### Summary

When we write an expression like $$2^n$$, we call $$n$$ the exponent.

If $$n$$ is a positive whole number, it tells how many factors of 2 we should multiply to find the value of the expression. For example, $$2^1=2$$, and $$2^5=2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2$$.

There are different ways to say $$2^5$$. We can say “two raised to the power of five” or “two to the fifth power” or just “two to the fifth.”